Epicycloid and Hypocycloid

Some epi/hypocycloids. The fraction on the upper coners are the two signed parameters b that generates the same curve. (see the Description section below.) Those that can only be generated as hypocycloids are colored red.

History

Cycloidal curves were first conceived by
Rømer (a Dane) in 1674 while studying the best form for gear teeth.
Galileo and
Mersenne had already (1599) discovered the ordinary
cycloid. The beautiful double generation theorem of these curves was first noticed by
Daniel Bernoulli in 1725. Astronomers find forms of the cycloidal curves in various coronas. They also occur as
caustics. The measuring of curve length was given by Newton in his
Principia.

Description

Epicycloid and hypocycloid both describe a family of curves. Epicycloid is a special case of epitrochoid, and hypocycloid is a special case of hypotrochoid. Specifically, epi/hypocycloid is the trace of a point on a circle rolling upon another circle without slipping. When a circle is rolling externally upon a fixed circle -- in the same manner a coin rolls around another -- we have epicycloid. When the rolling is internal, we have hypocycloid. The external and internal rolling can be distinguished as whether the circles have the same concavity at the point of contact. From another aspect, if the two circles are situated so that one is inside another, then it's a hypocycloid, otherwise its epicycloid. Note that when a circle is inside another, the rolling circle can be either the smaller circle or the larger one.

Various sizes of circles generates different epi/hypocycloids. Let the radius of the fixed circle be a, the radius of the rolling circle be b. Since the ratio a/b defines the shape of the curve, we will always let a:=1. Thus epi/hypocycloids are curves of one parameter b. The parametric formula for both curves can be easily derived from analytic geometry. It is: {(a + b)*Cos[t] + b*Cos[(a + b)/b*t], (a + b)*Sin[t] + b*Sin[(a + b)/b*t]}. This parametrization has the property that when b is positive, it generate a epicycloid. When b is negative, it is a hypocycloid. Thus, a signed parameter b uniquely defines a epi/hypocycloid.

Like epi/hypotrochoids, epi/hypocycloids have the double generation property: for any epi/hypocycloids, there are two unequal signed parameters b1, b2 that gives the same curve. (see Double Generation in the Properties section below.) However, curves that has cusps pointing towards the center are traditionally identified as epicycloids, even though they are also hypocycloids, and curves that has cusps pointing away from center are identified as hypocycloids. We define “n-cusped simple epi/hypocycloid” to be epi/hypocycloids with signed parameter b:=1/+-n. Such curves look like a simple loop, and the curve does not cross itself. They are often just called n-cusped epi/hypocycloid although there are infinity number of epi/hypocycloids with a given number of cusps.

We define the vertexes of the epicycloid to be points on the curve that coincides with a circumscribed circle. Similarly, the vertexes of a hypocycloid are the points on the curve that coincides with a inscribed circle.

Formulas

a is the radius of the fixed circle. b is the radius of the rolling circle. If b is positive, this formula gives epicycloid. If b is negative, this formula gives the hypocycloid. Let a:=1, b:=p/q where p, q are coprime. Then the formula has a period of 2*π*p. If b is irrational, the formula is nonperiodic.
varing b animation

The formula is easily derived using vectors. Starting with a point on circle a*{Cos[t],Sin[t]}.
Translate that to the center of the rolling circle we have a*{Cos[t],Sin[t]}+b*{Cos[t],Sin[t]}. Then we need another translation in the direction of the fixed tracing point relative to the rolling circle. To figure out which direction that is, we need to figure out how many revolutions there are for the rolling circle to go around the fixed circle. Some thinking and experiment shows it to be (a + b)/b.

Now, suppose the tracing point is fixed h distance on the rolling circle, then we have the formula for general epitrochoid and hypotrochoid:

Properties

Double Generation

Let the radius of the fixed circle be 1. The epicycloid with rolling circle radius b is equal to the hypocycloid with rolling circle radius b+1. This means that every epicycloid can be generated as a hypocycloid. A hypocycloid can be generated as a epiycloid if and only if its rolling circle is larger than the fixed circle. For hypocycloid whose rolling circle is smaller than the fixed circle, there are two sizes of the rolling circle that will generate the same curve. If b1 and b2 are the radiuses of such two rolling circles, then b1+b2==1.

Evolute

The evolute of a epi/hypocycloid is equal to itself. It is the same as itself scaled and rotated so that the old curve's cusp/vertex coincides with the new curve's vertex/cusp. The scaling factor is 1/(1+2*b), where b is the signed parameter.

Consecutive evolutes of a 6-cusped simple epicycloid (left) and a 12-cusped simple hypocycloid (right). The evolute of a epicycloid is smaller. For hypocycloid, larger.Epicycloids (blue) and hypocycloids (blue) with their evolutes (red).

Pedal and Radial

The pedal curve of a epi/hypocycloid (with signed parameter b) with respect to its center is its radial curve scaled (and reflected) by s:=-(1+2*b)^2/(4*b*(1+b)). For epicycloid, the pedal is larger. For hypocycloid, the radial is larger.

The evolute and radial of a 4-cusped simple epicycloid (left) and a 5-cusped simple hypocycloid (right).

Pedal, Radial, and Rose

The pedal of a epi/hypocycloid (with signed parameter b) with respect to its center is equal to the rose r==Cos[1/(-1-2*b)*theta]. The vertexes of epi/hypocycloids coincides with the roses' petal tips. Since pedal of a epi/hypocycloid with respect to its center is equal to its radial, therefore the radial of epi/hypocycloids are also roses.